Method for controlling an electric motor, control unit and electric motor

ABSTRACT

For ensuring high-precision control of a planar motor ( 1 ) comprising a magnet ( 2 ) and j coils ( 3 ), j=1 . . . N, wherein currents T 7  can flow through the coils ( 3 ) such that a force and a moment are generated that interact with the magnet ( 2 ), it has been proposed to determine force and moment needed to change the relative position of magnet ( 2 ) and coils from a present position to a desired position, and then to determine the currents T 7  necessary for generating this force and moment in the computing means ( 43 ) of the control unit ( 4 ) of the electric motor ( 1 ). The coil currents are then regulated accordingly with regulating means ( 44 ). The relative position of magnet ( 2 ) and coils ( 3 ) is measured with measuring means ( 5 ) and fed into the first input means ( 41 ) of the control unit ( 4 ).

The invention relates to a method for controlling an electric motor, particularly a planar motor, wherein a magnet is positioned with respect to j coils, j=1 . . . N, wherein currents I_(j) can flow through the coils such that a force and a moment are generated that interact with the magnet, a control unit for controlling an electric motor and an electric motor.

Electric motors are used in a variety of electrical equipment, especially equipment for high precision positioning. One field of application is for example the positioning of wafers during photolithography and other semiconductor processing with the help of linear or planar electric motors.

US 2003/0085676 A1 describes a system and method for independently controlling planar motors to move and position in six degrees of freedom. The electric planar motor comprises a moving magnet array and a coil array. The current supplied to the coils of the coils array interacts with the magnetic field of the magnets of the magnet array to generate forces between the magnet and coil arrays. The generated forces provide motion of the magnet array relative to the coil array in a first, second and third directions generally orthogonal to each other, as well as rotation about the first, second and third directions.

The method according to US 2003/0085676 A1 comprises the steps of determining the currents to be applied to the coils to generate forces between the magnet array and the coil array in a first, second, and third directions; determining the resultant torque about the first, second and third directions between the magnet array and the coil array generated by the forces generated by the determined currents; determining current adjustments to compensate for or cancel out the resultant torque; and applying a sum of the determined currents and determined current adjustments to the coils to interact with the magnetic fields of the magnetic array.

It is desirable to provide the possibility to control an electric motor, wherein a magnet is positioned with respect to one or more coils, wherein currents Ij, j=1 . . . N can flow through the coils such that a force and a moment are generated that interact with the magnet with high precision with respect to positioning and path to follow.

In a first aspect of the present invention, a method for controlling an electric motor, particularly a planar motor, wherein a magnet is positioned with respect to j coils, j=1 . . . N, wherein currents I_(j) can flow through the coils such that a force and a moment are generated that interact with the magnet, with the steps of determining the present relative position of magnet and coils; determining the force {right arrow over (F)}^(prescr) and moment {right arrow over (M)}^(prescr) needed to change the relative position of magnet and coils from the present position to a desired position; determining the necessary currents I_(j) ^(nec) for generating the force {right arrow over (F)}^(prescr) and moment {right arrow over (M)}^(prescr), wherein a further constraint concerning the system magnet-coils is taken into account for determining the currents I_(j) ^(nec); and applying the determined currents I_(j) ^(nec) to the j coils.

The fact of considering from the beginning both force and moment necessary for a desired relative movement between magnet and coils allows for a very accurate control of the movement. By also taking into account a further constraint concerning the system magnet-coils, the system as a whole is optimized, and it provides the possibility of determining unique currents, thus enhancing the accuracy of the control.

The method according to the invention is especially advantageous, if the number N of current Ij is larger than the number of degrees of freedom of the electric motor, leading to unique currents I_(j) ^(nec). The number of degrees of freedom is equivalent to the number of independent variables of {right arrow over (F)}^(prescr) and {right arrow over (M)}^(prescr).

In preferred embodiments, the further constraint taken into account is the minimization of total power dissipation of the electric motor, leading to an electric motor optimized with respect to efficiency, or the constraint of having a prescribed distribution of force and moment in space, which can lead to a minimal deformation of the magnet plate during motion.

In most preferred embodiments of the present invention, the method of Lagrange is utilized for determining the unique currents I_(j) ^(nec). A functional depending of currents I_(j) and Lagrange multipliers λ_(i) and taking into account the force and moment needed for changing the relative position between magnet and coils form the present position to a desired position as well as the chosen constraint is minimized, giving the currents I_(j) ^(nec).

Advantageously, the force and moment needed for changing the relative position between magnet and coils from the present position to a desired position are not individually determined after every new position through computing, but a set of different forces {right arrow over (F)}^(prescr) and moments {right arrow over (M)}^(prescr), each needed to change the relative position of magnet and coils from a present position to a desired position is computed beforehand and provided as database. This reduces the required computing resources and increases the reaction time during the control.

In a further aspect of the present invention, a control unit for controlling an electric motor, particularly a planar motor, wherein a magnet is positioned with respect to j coils, j=1 . . . N, wherein currents I_(j) can flow through the coils such that a force and a moment are generated that interact with the magnet, is provided with first input means to receive information on the present relative position of magnet and coils; second input means to receive information on force {right arrow over (F)}^(prescr) and moment {right arrow over (M)}^(prescr) needed to change the relative position of magnet and coils from the present position to a desired position; computing means for computing the necessary currents I_(j) ^(nec) for generating the force {right arrow over (F)}^(prescr) and moment {right arrow over (M)}^(prescr), wherein a further constraint the system magnet-coils is taken into account for defining I_(j) ^(nec); and regulating means for regulating the currents I_(j) to apply the computed currents I_(j) ^(nec) to the j coils.

The second input means of the control unit may be arranged as storing means for storing a set of forces {right arrow over (F)}^(prescr) and moments {right arrow over (M)}^(prescr), each needed to change the relative position of magnet and coils from a present position to a desired position, or may be arranged as computing means for computing the {right arrow over (F)}^(prescr) and moment {right arrow over (M)}^(prescr) needed to change the relative position of magnet and coils from the present position to a desired position.

In a last aspect of the invention, an electric motor is provided, comprising a magnet and j coils j=1 . . . N, wherein currents I_(j), can flow through the coils such that a force and a moment are generated that interact with the magnet, a control unit according to the invention.

In preferred embodiments, the motor is a planar motor, in most preferred embodiments a planar motor with six degrees of freedom.

It is possible to have the coils move with respect to the magnet or both the coils and the magnet moving. But it is preferred to have the magnet move with respect to the coils. This avoids hoses and cables impeding the free motion of the coils.

Advantageously, the motor has means for measuring the relative position of magnet and coils, thus improving the accuracy of control and positioning.

A detailed description of the invention is provided below. Said description is provided by way of a non-limiting example to be read with reference to the attached drawings in which:

FIG. 1 is a block diagram illustrating an embodiment of the control method according to the invention; and

FIG. 2 is a block diagram illustrating an embodiment of the control unit and electric motor according to the invention.

FIG. 1 shows a block diagram illustrating an embodiment of the method according to the present invention as control loop the motor shown in FIG. 2. The magnet plate has to move along a prescribed path with a prescribed velocity. In order to realize this motion, a force and moment have to be exerted on the magnet plate. This force and moment are generated by the interaction of the magnetic field from the magnet plate and the currents flowing through the coils. These currents can be controlled, which means that the force and moment acting at the plate can be controlled.

In order to generate a required force and moment, the currents through the coils have to be prescribed. If a force and moment act at the plate, the plate will move and attains a new position. This position is measured with respect to position {right arrow over (x)} and orientation {right arrow over (ω)} of the magnet plate. The control loop takes care that the force and moment and therefore the currents are prescribed such that the plate follows the prescribed path with the prescribed velocity.

The starting point, i.e. the first relative position of magnet and coils, is given as set point in step 101. The magnet has to move along a path to a position {right arrow over (x)} and orientation {right arrow over (ω)} with a prescribed velocity. The necessary force and moment that have to act on the magnet are determined in step 102. It is possible to look the values up in a predetermined database as well as to punctually compute the values. In the next step 103, the necessary currents Ij are determined with the constraints, that the determined force and moment have to be generated, and a further constraint, e.g. minimal power dissipation or a prescribed distribution of force and moment. Examples on how to determine the currents will be given below.

Once the currents have been determined, they are applied to the one or more coils, thus moving the magnet with respect to the coils to position {right arrow over (x)} and orientation {right arrow over (ω)} with the prescribed velocity (step 104). Then the present relative position and orientation are measured (step 105) and taken as starting point for a new loop.

In FIG. 2, the corresponding electric motor 1 is shown. It has a magnet 2 positioned over an array of coils 3. Magnet 2 and coils 3 are arranged in the x-y-plane, with the z-direction pointing from the coils 3 to the magnet 2. As can be seen, only thirty coils are at least partly covered by the magnet 2. To simplify the control of the electric motor, especially the determination of the necessary currents, it is possible to have currents flow only through the coils 3 at least partly covered by the magnet 2 and to take only these coils into account, when determining force, momentum and currents. Then a path has to be divided into sub-paths, each sub-path having a defined set of coils 3 at least partly covered by the magnet 2. It will be further noticed, that the magnet 2 has not to be a single magnet, but can also be an array of magnets. This only makes the prescribed force and moment more complex.

The electric motor 1 is controlled by control unit 4. Through a first input means 41, the computing means 43 gets information on present relative position of magnet 2 and coils 3. This information can be provided by a measuring means 5, e.g. based on optical measurements with lasers. This information can also stored as first set point, corresponding to the last present position. But accuracy is improved, if the present position is at least from time to time measured independently.

The information on the prescribed force and momentum is provided to the computing means 43 through the second input means 42. This can be, for example, a storage means, where a database containing a set of forces and moments needed for changing the relative position from a present position to a desired position. It can also be a further computing means for computing the actual force and moment, and then even be integrated into the computing means 43.

Having all the information needed, the computing means 43 can compute the necessary currents for generating the determined force and moment. Examples on how to do the computing will be given below. The regulating means 44 then regulates the coil currents such that the computed currents are applied, thus moving the magnet 2 with respect to the coils 3 to position {right arrow over (x)} and orientation {right arrow over (ω)} with the prescribed velocity.

Possible ways of determining currents I_(j) ^(nec) are explained in the following with respect to the example of a planar motor with a moving plate and a number of fixed coils and having six degrees of freedom.

Currents flow through the coils such that a force and moment are applied to the magnet. Each current I_(j) causes a force {right arrow over (F)}_(j) and moment {right arrow over (M)}_(j) acting on the magnet. Hence, the total force {right arrow over (F)} and moment {right arrow over (M)} acting on the magnet are

$\begin{matrix} {{\overset{}{F} = {\sum\limits_{j = 1}^{N}{\overset{}{F}}_{j}}},{\overset{}{M} = {\sum\limits_{j = 1}^{N}{\overset{}{M}}_{j}}},} & (1) \end{matrix}$

where N is the number of currents contributing to the exerted load on the magnet. The problem is to determine the currents I_(j), (j=1, . . . N) such that a prescribed force {right arrow over (F)}^(presc) and moment {right arrow over (M)}^(presc) are applied to the magnet.

Consider a unit current I_(j)=1. This unit current through the coil exerts a force {right arrow over (F)}_(j) ¹ and moment {right arrow over (M)}j¹ on the magnet. A current with value I_(j) will exert a force {right arrow over (F)}_(j)={right arrow over (F)}j¹I_(j) and moment {right arrow over (M)}_(j)={right arrow over (M)}_(j) ¹I_(j) (no summation over j). Superposition of all current I_(j), (j=1, . . . N) yields that the currents have to satisfy

$\begin{matrix} {\begin{pmatrix} {\overset{}{F}}^{presc} \\ {\overset{}{M}}^{presc} \end{pmatrix} = {{\sum\limits_{j = 1}^{N}{\begin{pmatrix} {\overset{}{F}}_{j}^{1} \\ {\overset{}{M}}_{j}^{1} \end{pmatrix}I_{j}}} = {\begin{pmatrix} {\overset{}{F}}_{1}^{1} & \ldots & {\overset{}{F}}_{N}^{1} \\ {\overset{}{M}}_{1}^{1} & \ldots & {\overset{}{M}}_{N}^{1} \end{pmatrix}\left( {\begin{matrix} I_{1} \\ \vdots \\ I_{N} \end{matrix}.} \right)}}} & (2) \end{matrix}$

To simplify the notations we introduce the vector {right arrow over (T)}^(presc)=({right arrow over (F)}^(presc),{right arrow over (M)}^(presc))^(T), the influence matrix

$F = \begin{pmatrix} {\overset{}{F}}_{1}^{1} & \ldots & {\overset{}{F}}_{N}^{1} \\ {\overset{}{M}}_{1}^{1} & \ldots & {\overset{}{M}}_{N}^{1} \end{pmatrix}$

and current vector {right arrow over (I)}=(I₁, . . . , I_(N))^(T). Then the constraint (2) is rewritten as

$\begin{matrix} {{\overset{}{T}}^{presc} = {{F\overset{}{I}\mspace{14mu} {or}\mspace{14mu} T_{i}^{presc}} = {\sum\limits_{j = 1}^{N}{F_{ij}{I_{j}\left( {i = {1\mspace{14mu} \ldots \mspace{14mu} 6}} \right)}}}}} & (3) \end{matrix}$

If N=6 and the matrix of influence factors is not singular, the currents follow from (2) and are

{right arrow over (I)}=F⁻¹{right arrow over (T)}^(presc).  (4)

If N<6 and the rank of the matrix of influence factors F equals N, only unique currents I_(j) (j=1 . . . N) exist, if the prescribed force and moment ({right arrow over (F)}^(presc),{right arrow over (M)}^(presc))^(T) are in the space spanned by ({right arrow over (F)}_(j) ¹,{right arrow over (M)}_(j) ¹)^(T) (j=1 . . . N). These currents are determined by the least squares solution of (3)

{right arrow over (I)}=(F ^(T) F)⁻ F ^(T) {right arrow over (T)} ^(presc).  (5)

If the prescribed force and moment are not in the space spanned by ({right arrow over (F)}_(j) ¹,{right arrow over (M)}_(j) ¹)^(T) (j=1 . . . N) no combination of currents exist that can generate the prescribed force and moment.

If N<6, the rank of the matrix of influence factors is smaller than N and if ({right arrow over (F)}^(presc),{right arrow over (M)}^(presc))^(T) is in the space spanned by ({right arrow over (F)}_(j) ¹,{right arrow over (M)}_(j) ¹)^(T) (j=1 . . . N), the currents I_(j) (j=1 . . . N) to generate the prescribed force {right arrow over (F)}^(presc) and moment {right arrow over (M)}^(presc) are not unique. In this case and in the case that N>6 additional demands have to be imposed on the currents to obtain unique values I_(j) to generate the prescribed force and moment.

Generally speaking, the required force and moment represent 6 constraints concordant with the six degrees of freedom of the motor. In general, the number N of coils that influence the force and moment acting at the plate is greater than 6. A typical number for the number of currents is between 20 and 30, e.g. 27. Then the problem is how to prescribe the currents in order that a required force and moment on the plate are generated. Hence, N>6 variables have to be determined, such that 6 constraints are satisfied.

The currents are the solution of an optimization/minimization problem with the constraint that the required force {right arrow over (F)}^(required) and moment {right arrow over (M)}^(required) are generated. Hence, if one uses the method of Lagrange, the currents are the variables at which the functional

$\begin{matrix} {{J\left( {I_{1},\ldots \mspace{14mu},I_{N},\lambda_{1},\ldots \mspace{14mu},\lambda_{6}} \right)} = {{G\left( {I_{1},\ldots \mspace{14mu},I_{N}} \right)} - {\sum\limits_{i = 1}^{6}{\lambda_{i}\left( {T_{i}^{required} - {T_{i}\left( {I_{1},\ldots \mspace{14mu},I_{N}} \right)}} \right)}}}} & (6) \end{matrix}$

has its minimum. The index i in the expression for J indicates components of the vectors {right arrow over (T)}

$\left( {\overset{}{T} = \begin{pmatrix} \overset{}{F} \\ \overset{}{M} \end{pmatrix}} \right)$

and {right arrow over (λ)}, λ_(1, . . . , 6) being Lagrange multipliers. The function {right arrow over (T)}(I₁, . . . , I_(N)) defines the relation between the currents I₁, . . . , I_(N) and the generated force {right arrow over (P)} and moment {right arrow over (M)}. In this case the force and moment are linear dependent on the currents, hence {right arrow over (T)}=A({right arrow over (x)},{right arrow over (ω)}){right arrow over (I)}, where the tensor A({right arrow over (x)},{right arrow over (ω)}) depends on the position {right arrow over (x)} and the orientation {right arrow over (ω)} of the magnet plate with respect to the fixed coils. The function G(I₁, . . . , I_(N)) defines the function to be minimized, with the constraint that the required force and moment

$\left( {{\overset{}{T}}^{required} = \begin{pmatrix} {\overset{}{F}}^{required} \\ {\overset{}{M}}^{required} \end{pmatrix}} \right)$

are generated, and is equivalent to the further constraint to be taken into account for determining I_(j) ^(necc). This function G(I₁, . . . , I_(N)) has to be chosen. Then, the condition that the function J has to be minimal generates sufficient extra conditions to determine uniquely the currents I₁, . . . , I_(N) (and it determines uniquely the extra introduced so called Lagrange multipliers λ₁, . . . , λ₆). Hence, unique currents I₁, . . . , I_(N) are determined that minimize some function G(I₁, . . . , I_(N)) with the constraint that the required force {right arrow over (F)}^(required) and moment {right arrow over (M)}^(required) are generated.

1. Minimal Power Dissipation

The function G(I₁, . . . , I_(N)) to be optimized has to be chosen. A suitable choice is the total power dissipation in the coils, hence

$\begin{matrix} {{G\left( {I_{1},\ldots \mspace{14mu},I_{N}} \right)} = {\sum\limits_{i = 1}^{N}{R_{i}I_{i}^{2}}}} & (7) \end{matrix}$

where the R_(i) are the resistances of the coils and the power dissipation caused by a current I_(j) is P_(j)=R_(j)I_(j) ² (no summation). If N>6 an additional demand on the currents I_(j) can be that the total power dissipation of the currents is minimal.

Next, the currents through the coils are determined by minimizing the total power dissipation with the constraint (3), hence the currents are determined by minimizing the functional

$\begin{matrix} {{{J\left( {I_{1},\ldots \mspace{14mu},I_{N}} \right)} = {{\sum\limits_{j = 1}^{N}{R_{j}I_{j}^{2}}} - {\sum\limits_{i = 1}^{6}{\lambda_{i}\left( {T_{i} - {\sum\limits_{j = 1}^{N}{F_{ij}I_{j}}}} \right)}}}},} & (8) \end{matrix}$

where λ_(i) are Lagrange multipliers.

This functional has a minimum if

$\begin{matrix} {{\frac{\partial J}{\partial I_{k}} = {{{2R_{k}I_{k}} + {\sum\limits_{i = 1}^{6}{\lambda_{i}F_{ik}}}} = {0\left( {k = {1\mspace{14mu} \ldots \mspace{14mu} N}} \right)}}}{and}} & (9) \\ {T_{i}^{presc} = {\sum\limits_{j = 1}^{N}{F_{ij}{{I_{j}\left( {i = {1\mspace{14mu} \ldots \mspace{14mu} 6}} \right)}.}}}} & (3) \end{matrix}$

To simplify the solution of the equations (3) and (9), the currents are written as

$\begin{matrix} {I_{k} = {{\sum\limits_{i = 1}^{6}{\lambda_{i}{I_{ki}\left( {k = {1\mspace{14mu} \ldots \mspace{14mu} N}} \right)}\mspace{14mu} {or}\mspace{14mu} \overset{}{I}}} = {I\; {\overset{}{\lambda}.}}}} & (10) \end{matrix}$

Substitution of (10) in (9) yields

$\begin{matrix} \begin{matrix} {\frac{\partial J}{\partial I_{k}} = {{2R_{k}{\sum\limits_{i = 1}^{6}{I_{ki}\lambda_{i}}}} + {\sum\limits_{i = 1}^{6}{\lambda_{i}F_{ik}}}}} \\ {= {\sum\limits_{i = 1}^{6}{\left( {{2R_{k}I_{ki}} + F_{ik}} \right)\lambda_{i}}}} \\ {= {0\left( {k = {1\mspace{14mu} \ldots \mspace{14mu} N}} \right)}} \end{matrix} & (11) \end{matrix}$

Hence solving (11) yields

$\begin{matrix} {I_{ik} = {{- \frac{F_{ik}}{2R_{k}}}\left( {i = {1\mspace{14mu} \ldots \mspace{14mu} 6}} \right)\left( {k = {1\mspace{14mu} \ldots \mspace{14mu} N}} \right)}} & (12) \end{matrix}$

The Lagrange multipliers λ_(i) follow from the constraint (3). Substitution of (10) and (12) in (3) yields

$\begin{matrix} \begin{matrix} {T_{i}^{presc} = {\sum\limits_{j = 1}^{N}{F_{ij}I_{j}}}} \\ {= {\sum\limits_{j = 1}^{N}{F_{ij}I_{j}}}} \\ {= {\sum\limits_{j = 1}^{N}{F_{ij}{\sum\limits_{k = 1}^{6}{\lambda_{k}I_{jk}}}}}} \\ {= {\sum\limits_{k = 1}^{6}{\lambda_{k}{\sum\limits_{j = 1}^{N}{F_{ij}I_{jk}}}}}} \\ {= {\sum\limits_{k = 1}^{6}{\lambda_{k}{\sum\limits_{j = 1}^{N}\frac{{- F_{ij}}F_{kj}}{2R_{j}}}}}} \end{matrix} & (13) \\ {{or}{{T_{i}^{presc} = {\sum\limits_{k = 1}^{6}{A_{ik}\lambda_{k}}}},{{\left( {i = {1\mspace{14mu} \ldots \mspace{14mu} 6}} \right)\mspace{14mu} {with}\mspace{14mu} A_{ik}} = {\sum\limits_{j = 1}^{N}\frac{{- F_{ij}}F_{kj}}{2R_{j}}}}}{or}{{\overset{}{T}}^{presc} = {A\; {\overset{}{\lambda}.}}}} & \; \end{matrix}$

Hence the Lagrange multipliers have to be solved from (13) and are given by

{right arrow over (λ)}=A⁻¹{right arrow over (T)}^(presc).  (14)

To obtain the currents I_(j), (12) and (14) are substituted in (10), then

$\begin{matrix} {I_{k} = {{\sum\limits_{i = 1}^{6}{\lambda_{i}I_{ki}}} = {\sum\limits_{i = 1}^{6}{\lambda_{i}\frac{- F_{ik}}{2R_{k}}{\left( {k = {1\mspace{14mu} \ldots \mspace{14mu} N}} \right).}}}}} & (15) \end{matrix}$

Hence, the currents given by (15) are the currents that deliver the prescribed force {right arrow over (F)}^(presc) and moment {right arrow over (M)}^(presc) with minimal power dissipation.

As example, consider the case that all resistances of the coils are equal, hence R_(j)=R (j=1 . . . N). Then the matrices A (13) and I (12) are

$\begin{matrix} {{A = {\frac{- 1}{2R}{FF}^{T}}},{I = {\frac{- 1}{2R}F^{T}}}} & \left( {E{.1}} \right) \end{matrix}$

and the currents are given by

$\begin{matrix} \begin{matrix} {\overset{}{I} = {I\; A^{- 1}{\overset{}{T}}^{presc}}} \\ {= {\frac{- 1}{2R}{F^{T}\left( {\frac{- 1}{2R}{FF}^{T}} \right)}^{- 1}{\overset{}{T}}^{presc}}} \\ {= {{F^{T}\left( {FF}^{T} \right)}^{- 1}{{\overset{}{T}}^{presc}.}}} \end{matrix} & \left( {E{.2}} \right) \end{matrix}$

Hence, as expected the currents are independent of the resistance of the coils.

2. Prescribed Force and Moment Distribution

Another requirement can be a desired force and moment per unit area distribution over the plate, hence a desired distribution {right arrow over (T)}_(dis) ^(desired)({right arrow over (x)}). Then the function G(I₁, . . . I_(N)) becomes

$\begin{matrix} {{{G\left( {I_{1},\ldots \mspace{14mu},I_{N}} \right)} = {\int{\int_{D}{{\left( {{{\overset{}{T}}_{dis}^{desired}\left( \overset{}{x} \right)} - {{\overset{}{T}}_{dis}\left( \overset{}{x} \right)}} \right) \cdot \left( {{{\overset{}{T}}_{dis}^{desired}\left( \overset{}{x} \right)} - {{\overset{}{T}}_{dis}\left( \overset{}{x} \right)}} \right)}{a}}}}},} & (16) \end{matrix}$

where the integration has to be carried out over the area D of the plate and {right arrow over (T)}_(dis)({right arrow over (x)}) represents the generated force and moment per unit area of the plate at a location {right arrow over (x)} of the plate. For practical reasons in general a numerical method (e.g. discretization) is used to calculate the surface integral. The area D of the plate can be divided into M sub-areas D_(k), then the integral over D can be approximated by

$\begin{matrix} {{G\left( {I_{1},\ldots \mspace{14mu},I_{N}} \right)} \approx {\sum\limits_{k = 1}^{M}{\left( {{\overset{\rightarrow}{T}}_{{dis},k}^{desired} - {\overset{\rightarrow}{T}}_{{dis},k}} \right) \cdot \left( {{\overset{\rightarrow}{T}}_{{dis},k}^{desired} - {\overset{\rightarrow}{T}}_{{dis},k}} \right)}}} & (17) \end{matrix}$

where {right arrow over (T)}_(dis,k) ^(desired) represents the desired force and moment acting at the part D_(k) of the total area D of the magnet plate and {right arrow over (T)}_(dis,k)=A^(k){right arrow over (I)} represents the force and moment acting at D_(k) due to the currents through the coils. The total force and moment are given by

$\begin{matrix} {{\overset{\rightarrow}{T}}^{total} = {{\sum\limits_{k = 1}^{M}{\overset{\rightarrow}{T}}_{{dis},k}} = {{\sum\limits_{k = 1}^{M}{A^{k}\overset{\rightarrow}{I}}} = {A{\overset{\rightarrow}{I}.}}}}} & (18) \end{matrix}$

This total force and moment have to be equal to the required force and moment {right arrow over (T)}^(required) acting at the magnet plate. Hence, the functional to be optimized becomes

$\begin{matrix} {{J\begin{pmatrix} {I_{1},\ldots \mspace{14mu},I_{N},} \\ {\lambda_{1},\ldots \mspace{14mu},\lambda_{6}} \end{pmatrix}} = {{\sum\limits_{k = 1}^{M}\begin{Bmatrix} {\left( {{\overset{\rightarrow}{T}}_{{dos},k}^{desired} - {A^{k}I}} \right) \cdot} \\ \left( {{\overset{\rightarrow}{T}}_{{dis},k}^{desired} - {A^{k}I}} \right) \end{Bmatrix}} - {\overset{\rightarrow}{\lambda} \cdot \left( {{\overset{\rightarrow}{T}}^{required} - {AI}} \right)}}} & (19) \end{matrix}$

More in detail, one has to divide the magnet plate into a number of area elements. These area elements are not necessarily equal. Consider a unit current I_(j)=1. This unit current through the coil exerts a force {right arrow over (A)}_(j) ^(k) and moment {right arrow over (B)}_(j) ^(k) on area element k, if this area element is considered to be decoupled from all other area elements. A current with value I_(j) will exert a force {right arrow over (K)}_(j) ^(k)={right arrow over (A)}_(j) ^(k)I_(j) and moment {right arrow over (L)}_(j) ^(k)={right arrow over (B)}_(j) ^(k)I_(j) (no summation over j) on area element k. Superposition of all current I_(j), (j=1 . . . N) yields the following force {right arrow over (F)}^(k) and moment {right arrow over (M)}^(k) on area element k

{right arrow over (F)}^(k)=A^(k){right arrow over (I)},{right arrow over (M)}^(k)=B^(k){right arrow over (I)},k=1, . . . , M,  (20)

where M is the number of area elements. The total force and moment acting at the magnet plate follows from the coupling of all area elements, hence the total force {right arrow over (F)}^(tot) and total moment {right arrow over (M)}^(tot) are

$\begin{matrix} \begin{matrix} {{\overset{\rightarrow}{F}}^{tot} = {\sum\limits_{k = 1}^{M}{\overset{\rightarrow}{F}}^{k}}} \\ {= {\sum\limits_{k = 1}^{M}{A^{k}\overset{\rightarrow}{I}{\overset{\rightarrow}{M}}^{tot}}}} \\ {= {\sum\limits_{k = 1}^{M}\left( {{\overset{\rightarrow}{M}}^{k} + {{\overset{\rightarrow}{x}}^{k} \times {\overset{\rightarrow}{F}}^{k}}} \right)}} \\ {= {\sum\limits_{k = 1}^{M}\left( {{B^{k}\overset{\rightarrow}{I}} + {{\overset{\rightarrow}{x}}^{k} \times A^{k}\overset{\rightarrow}{I}}} \right)}} \end{matrix} & (21) \end{matrix}$

where {right arrow over (x)}^(k) is the vector from the point at which the total force and moment act to the location on the area element at which the force {right arrow over (F)}^(k) and moment {right arrow over (M)}^(k) act.

The distribution of force and moment over the magnet plate can be prescribed. If the magnet plate is discretized in area elements, this means that the distribution of force {right arrow over (F)}^(k) and moment {right arrow over (M)}^(k) acting on the area elements can be prescribed. Then the problem is to determine the currents trough the coils which deliver (approximately) these force and moment distributions. However, the currents through the coils have to generate a prescribed total force and total moment acting on the magnet plate. Hence, the following optimization problem has to be solved.

Determine the currents {right arrow over (I)} such that the prescribed distribution of forces {right arrow over (F)}^(k) and moments {right arrow over (M)}^(k) are satisfied in a least squares sense, with the constraint that the required total force {right arrow over (F)}^(tot) and moment {right arrow over (M)}^(tot) are generated. Hence the currents follow from minimizing the functional

$\begin{matrix} {{{J\left( {\overset{\rightarrow}{I},\overset{\rightarrow}{\lambda},\overset{\rightarrow}{\mu}} \right)} = {{\sum\limits_{k = 1}^{M}\begin{Bmatrix} {{\left( {{\overset{\rightarrow}{F}}^{k} - {A^{k}\overset{\rightarrow}{I}}} \right) \cdot \left( {{\overset{\rightarrow}{F}}^{k} - {A^{k}\overset{\rightarrow}{I}}} \right)} +} \\ {\left( {{\overset{\rightarrow}{M}}^{k} - {B^{k}\overset{\rightarrow}{I}}} \right) \cdot \left( {{\overset{\rightarrow}{M}}^{k} - {B^{k}\overset{\rightarrow}{I}}} \right)} \end{Bmatrix}} - {\overset{\rightarrow}{\lambda} \cdot \left( {{\overset{\rightarrow}{F}}^{tot} - {\sum\limits_{k = 1}^{M}{A^{k}\overset{\rightarrow}{I}}}} \right)} - {\overset{\rightarrow}{\mu} \cdot \left( {{\overset{\rightarrow}{M}}^{tot} - {\sum\limits_{k = 1}^{M}\left( {{B^{k}\overset{\rightarrow}{I}} + {{\overset{\rightarrow}{x}}^{k} \times A^{k}\overset{=}{I}}} \right)}} \right)}}},} & (22) \end{matrix}$

where {right arrow over (λ)} and {right arrow over (μ)} are Lagrange multipliers.

In order to determine the conditions for which the functional (22) has a minimum, we write it as a function of the components of the vectors {right arrow over (I)}, {right arrow over (λ)} and {right arrow over (μ)}, hence

$\begin{matrix} {{J\left( {I_{i},\lambda_{j},\mu_{k}} \right)} = {{\sum\limits_{m = 1}^{M}\begin{Bmatrix} {{\left( {{\overset{\rightarrow}{F}}_{n}^{m} - {A_{np}^{m}I_{p}}} \right) \cdot \left( {{\overset{\rightarrow}{F}}_{n}^{m} - {A_{np}^{m}I_{p}}} \right)} +} \\ {\left( {M_{n}^{m} - {B_{np}^{m}I_{p}}} \right) \cdot \left( {M_{n}^{m} - {B_{np}^{m}I_{p}}} \right)} \end{Bmatrix}} - {\lambda_{r} \cdot \left( {{\overset{\rightarrow}{F}}_{r}^{tot} - {\sum\limits_{m = 1}^{M}{A_{rp}^{m}I_{p}}}} \right)} - {\mu_{s} \cdot \left( {M_{s}^{tot} - {\sum\limits_{m = 1}^{M}\left( {{B_{sp}^{m}I_{p}} + {ɛ_{sqt}x_{q}^{m}A_{tp}^{m}I_{p}}} \right)}} \right)}}} & (23) \end{matrix}$

where

ε_(sqt)=1, if sqt is 123, 312, 231,

ε_(sqt)=−1, if sqt is 132, 321, 213,

ε_(sqt)=0, otherwise.

Note, that in the expression of the function (23), the Einstein summation convention is used for the indices n, p, q, r, s and t (i.e. summation over these indices if they appear twice in a term). Differentiation with respect to I_(i), λ_(j) and μ_(k) and setting the result to zero, yields

$\begin{matrix} {0 = {\frac{\partial J}{\partial I_{i}} = {{\sum\limits_{m = 1}^{M}\begin{Bmatrix} {{{- 2}\left( {{\overset{\rightarrow}{F}}_{n}^{m} - {A_{np}^{m}I_{p}}} \right)A_{ni}^{m}} -} \\ {2\left( {M_{n}^{m} - {B_{np}^{m}I_{p}}} \right)B_{ni}^{m}} \end{Bmatrix}} - {\lambda_{r}{\sum\limits_{m = 1}^{M}A_{ri}^{m}}} - {\mu_{s}{\sum\limits_{m = 1}^{M}\left( {B_{si}^{m} + {ɛ_{sqt}x_{q}^{m}A_{ti}^{m}}} \right)}}}}} & (24) \\ {{0 = {\frac{\partial J}{\partial I_{i}} = {{{- 2}{\sum\limits_{m = 1}^{M}\left( {{A_{ni}^{m}{\overset{\rightarrow}{F}}_{n}^{m}} + {B_{ni}^{m}M_{n}^{m}}} \right)}} + {2{\sum\limits_{m = 1}^{M}{\left( {{A_{ni}^{m}A_{np}^{m}} + {B_{ni}^{m}B_{np}^{m}}} \right)I_{p}}}} + {\sum\limits_{m = 1}^{M}{A_{ri}^{m}\lambda_{r}}} + {\sum\limits_{m = 1}^{M}{\left( {B_{si}^{m} + {ɛ_{sqt}x_{q}^{m}A_{ti}^{m}}} \right)\mu_{s}}}}}},} & (25) \\ {\mspace{79mu} {{0 = {\frac{\partial J}{\partial\lambda_{j}} = {- \left( {F_{j}^{tot} - {\sum\limits_{m = 1}^{M}{A_{jp}^{m}I_{p}}}} \right)}}},}} & (26) \\ {\mspace{79mu} {0 = {\frac{\partial J}{\partial\mu_{k}} = {- {\left( {M_{k}^{tot} - {\sum\limits_{m = 1}^{M}\left( {{B_{kp}^{m}I_{p}} + {ɛ_{kqt}x_{q}^{m}A_{tp}^{m}I_{p}}} \right)}} \right).}}}}} & (27) \end{matrix}$

The functional (22) has a minimum, if (25), (26) and (27) are fulfilled. In vector and matrix notation these expressions become

$\begin{matrix} {{\overset{\rightarrow}{0} = {{{- 2}{\sum\limits_{k = 1}^{M}\left\{ {{\left( A^{k} \right)^{T}{\overset{\rightarrow}{F}}^{k}} + {\left( B^{k} \right)^{T}{\overset{\rightarrow}{M}}^{k}}} \right\}}} + {2{\sum\limits_{k = 1}^{M}{\left\{ {{\left( A^{k} \right)^{T}A^{k}} + {\left( B^{k} \right)^{T}B^{k}}} \right\} \overset{\rightarrow}{I}}}} + {\sum\limits_{k = 1}^{M}{\left( A^{k} \right)^{T} \cdot \overset{\rightarrow}{\lambda}}} + {\sum\limits_{k = 1}^{M}{\left\{ {\left( B^{k} \right)^{T} + \left( D^{k} \right)^{T}} \right\} \cdot \overset{\rightarrow}{\mu}}}}},} & (28) \\ {\mspace{79mu} {{\overset{\rightarrow}{0} = {{- {\overset{\rightarrow}{F}}^{tot}} + {\sum\limits_{k = 1}^{M}{A^{k}\overset{\rightarrow}{I}}}}},}} & (29) \\ {\mspace{79mu} {{\overset{\rightarrow}{0} = {{- {\overset{\rightarrow}{M}}^{tot}} + {\sum\limits_{k = 1}^{M}{\left\{ {B^{k} + D^{k}} \right\} \overset{\rightarrow}{I}}}}},}} & (30) \end{matrix}$

where the components of matrix D^(k) are D_(ij) ^(k)=ε_(ipq)x_(p) ^(k)A_(qj) ^(k), with

ε_(ipq)=1, if ipq is 123, 312, 231,

ε_(ipq)=−1, if ipq is 132, 321, 213,  (31)

ε_(ipq)=0 otherwise.

Next the following definitions are introduced

${A = {\sum\limits_{k = 1}^{M}A^{k}}},{B = {\sum\limits_{k = 1}^{M}B^{k}}},{C = {2{\sum\limits_{k = 1}^{M}\left\{ {{\left( A^{k} \right)^{T}A^{k}} + {\left( B^{k} \right)^{T}B^{k}}} \right\}}}},{D = {\sum\limits_{k = 1}^{M}D^{k}}},{\overset{\rightarrow}{E} = {2{\sum\limits_{k = 1}^{M}\left\{ {{\left( A^{k} \right)^{T}{\overset{\rightarrow}{F}}^{k}} + {\left( B^{k} \right)^{T}{\overset{\rightarrow}{M}}^{k}}} \right\}}}}$

Then the conditions (28)-(30) for a minimum of the functional (22) become

C{right arrow over (I)}={right arrow over (E)}−A ^(T){right arrow over (λ)}−(B+D)^(T){right arrow over (μ)}  (32)

{right arrow over (F)}^(tot)=A{right arrow over (I)}  (33)

{right arrow over (M)} ^(tot)=(B+D){right arrow over (I)}  (34)

From (32) it follows that

{right arrow over (I)}=C ⁻¹ {right arrow over (E)}−C ⁻¹ A ^(T){right arrow over (λ)}−C ⁻¹(B+D)^(T){right arrow over (μ)}  (35)

that can be rewritten as

$\begin{matrix} {\overset{\rightarrow}{I} = {{C^{- 1}\overset{\rightarrow}{E}} - {{C^{- 1}\begin{pmatrix} A \\ {B + D} \end{pmatrix}}^{T}{\begin{pmatrix} \overset{\rightarrow}{\lambda} \\ \overset{\rightarrow}{\mu} \end{pmatrix}.}}}} & (36) \end{matrix}$

Substitution of (35) in (33) and (34) yields, respectively,

{right arrow over (F)} ^(tot) =AC ⁻¹ {right arrow over (E)}−AC ⁻¹ A ^(T) {right arrow over (λ)}−AC ⁻¹(B+D)^(T){right arrow over (μ)},  (37)

{right arrow over (M)} ^(tot)=(B+D)C ⁻¹ {right arrow over (E)}−(B+D)C ⁻¹ A ^(T){right arrow over (λ)}−(B+D)C ⁻¹(B+D)^(T){right arrow over (λ)}.  (38)

Combining (37) and (38) gives the following set of linear equations for the Lagrange multipliers {right arrow over (λ)} and {right arrow over (μ)}

$\begin{matrix} {{\begin{pmatrix} {A\; C^{- 1}A^{T}} & {A\; {C^{- 1}\left( {B + D} \right)}^{T}} \\ {\left( {B + D} \right)C^{- 1}A^{T}} & {\left( {B + D} \right){C^{- 1}\left( {B + D} \right)}^{T}} \end{pmatrix}\begin{pmatrix} \overset{\rightarrow}{\lambda} \\ \overset{\rightarrow}{\mu} \end{pmatrix}} = {{\begin{pmatrix} A \\ {B + D} \end{pmatrix}C^{- 1}\overset{\rightarrow}{E}} - \begin{pmatrix} {\overset{\rightarrow}{F}}^{tot} \\ {\overset{\rightarrow}{M}}^{tot} \end{pmatrix}}} & (39) \end{matrix}$

Hence the solution for {right arrow over (λ)} and {right arrow over (μ)} is

$\begin{matrix} {\begin{pmatrix} \overset{\rightarrow}{\lambda} \\ \overset{\rightarrow}{\mu} \end{pmatrix} = {\begin{pmatrix} {A\; C^{- 1}A^{T}} & {A\; {C^{- 1}\left( {B + D} \right)}^{T}} \\ {\left( {B + D} \right)C^{- 1}A^{T}} & {\left( {B + D} \right){C^{- 1}\left( {B + D} \right)}^{T}} \end{pmatrix}^{- 1}\left\{ {{\begin{pmatrix} A \\ {B + D} \end{pmatrix}C^{- 1}\overset{}{E}} - \begin{pmatrix} {\overset{\rightarrow}{F}}^{tot} \\ {\overset{\rightarrow}{M}}^{tot} \end{pmatrix}} \right\}}} & (40) \end{matrix}$

Substitution of (40) into equation (36) gives the closed expression for the currents, directly as a function of prescribed force and moment.

The prescribed force distribution and moment distribution can be, for example, uniform. In the case that the total force and moment are prescribed with the condition of a uniform force and moment distribution on the area elements and with the condition that the area elements are equal in size, the force {right arrow over (F)}^(k) and moment {right arrow over (M)}^(k) acting at a surface element k are

$\begin{matrix} {{{\overset{\rightarrow}{F}}^{k} = {\frac{1}{M}{\overset{\rightarrow}{F}}^{tot}}},{{\overset{\rightarrow}{M}}^{k} = {\frac{1}{M}{\left( {{\overset{\rightarrow}{M}}^{tot} - {\sum\limits_{i = 1}^{M}{{\overset{\rightarrow}{x}}^{i} \times {\overset{\rightarrow}{F}}^{i}}}} \right).}}}} & \left( {E{.3}} \right) \end{matrix}$

In the case that the area elements are not equal and have size O^(k), with

$O = {\sum\limits_{k = 1}^{M}O^{k}}$

the total area of the magnet plate, the forces {right arrow over (F)}^(k) and moments {right arrow over (M)}^(k) belonging to a uniform distribution are

$\begin{matrix} {{{\overset{\rightarrow}{F}}^{k} = {\frac{O^{k}}{O}{\overset{\rightarrow}{F}}^{tot}}},{{\overset{\rightarrow}{M}}^{k} = {\frac{O^{k}}{O}{\left( {{\overset{\rightarrow}{M}}^{tot} - {\sum\limits_{i = 1}^{M}{{\overset{\rightarrow}{x}}^{i} \times {\overset{\rightarrow}{F}}^{i}}}} \right).}}}} & \left( {E{.4}} \right) \end{matrix}$

Although having described several preferred embodiments of the invention, those skilled in the art would appreciate that various changes, alterations, and substitutions can be made without departing from the spirit and concepts of the present invention. The invention is, therefore, claimed in any of its forms or modifications with the proper scope of the appended claims. For example various combinations of the features of the following dependent claims could be made with the features of the independent claim without departing from the scope of the present invention. Furthermore, any reference numerals in the claims shall not be construed as limiting scope.

LIST OF REFERENCE NUMERALS

-   1 electric motor -   2 magnet -   3 coil -   4 control unit -   5 measuring means -   41 first input means -   42 second input means -   43 computing means -   44 regulating means -   101 step of specifying set point -   102 step of determining force/moment -   103 step of determining currents -   104 step of moving magnet -   105 step of measuring position 

1. A method for controlling an electric motor, particularly a planar motor, wherein a magnet is positioned with respect to j coils, j=1 . . . N, wherein currents I_(j) can flow through the coils such that a force and a moment are generated that interact with the magnet, with the steps of: determining the present relative position of magnet and coils; determining the force {right arrow over (F)}^(prescr) and moment {right arrow over (M)}^(prescr) needed to change the relative position of magnet and coils from the present position to a desired position; determining the necessary currents I_(j) ^(necc) for generating the force {right arrow over (F)}^(prescr) and moment {right arrow over (M)}^(prescr), wherein a further constraint concerning the system magnet-coil is taken into account for defining the currents I_(j) ^(nec); applying the determined currents I_(j) ^(nec) to the j coils.
 2. The method according to claim 1, wherein the number N of currents I_(j) is larger than the number of degrees of freedom of the electric motor.
 3. The method according to claim 1, wherein the further constraint is minimal power dissipation.
 4. The method according to claim 1, wherein j>1 and the further constraint is a prescribed distribution of force and moment.
 5. The method according to claim 1, wherein a Lagrange functional is minimized for determining the currents I_(j) ^(nec).
 6. The method according to claim 1, wherein a set of different forces {right arrow over (F)}^(prescr) and moments {right arrow over (M)}^(prescr), each needed to change the relative position of magnet and coils from a present position to a desired position, is determined.
 7. A control unit (4) for controlling an electric motor (1), particularly a planar motor, wherein a magnet (2) is positioned with respect to j coils (3), j=1 . . . N, wherein currents I_(j) can flow through the coils (3) such that a force and a moment are generated that interact with the magnet (2), with first input means (41) to receive information on the present relative position of magnet (2) and coils (3); second input means (42) to receive information on force {right arrow over (F)}prescr and moment {right arrow over (M)}^(prescr) needed to change the relative position of magnet (2) and coils (3) from the present position to a desired position; computing means (43) for computing the necessary currents I_(j) ^(nec) for generating the force {right arrow over (F)}^(prescr) and moment {right arrow over (M)}^(prescr), wherein a further constraint concerning the system magnet-coil is taken into account for determining the currents I_(j) ^(nec); regulating means (44) for regulating the currents I_(j) to apply the computed currents I_(j) ^(nec) to the j coils (3).
 8. The control unit according to claim 7 wherein the second input means (42) is arranged as storing means for storing a set of forces {right arrow over (F)}^(prescr) and moments {right arrow over (M)}^(prescr), each needed to change the relative position of magnet (2) and coils (3) from a present position to a desired position.
 9. The control unit according to claim 7, wherein the second input means (42) is arranged as computing means for computing the {right arrow over (F)}^(prescr) and moment {right arrow over (M)}^(prescr) needed to change the relative position of magnet (2) and coils (3) from the present position to a desired position.
 10. An electric motor (1), comprising a magnet (2) and j coils (3), j=1 . . . N, wherein currents I_(j) can flow through the coils (3) such that a force and a moment are generated that interact with the magnet (2), a control unit (4) according to claim
 7. 11. The electric motor according to claim 10, wherein the motor (1) is a planar motor.
 12. The electric motor according to claim 10, wherein the motor (1) is a planar motor with six degrees of freedom.
 13. The electric motor according to claim 10, wherein the magnet (2) is movable with respect to the coils (3).
 14. The electric motor according to claim 10, having means (5) for measuring the relative position of magnet (2) and coils (3). 